Among all the variations of poker,
for draw poker (or classical poker) the probabilities involved are the
hardest to compute. If the odds for your own hand were partially figured
out before, the odds for opponents' hands were not. Players were only
able to figure out odds for opponents' hands in particular gaming
situations, by applying some basic combinatorial calculus and algebraic
cleverness. No formulas to cover all gaming situations and return
numerical values when inputing parameters were figured out before, due to
the assumed heaviness of such formulas.
Well,
these formulas have been finally worked out! They are the outcome of one
year of study, math work and tests. We found the right probability model
in which to apply the theory and quantify the card distributions in order
to work out the draw poker
probability formulas. They were built with an
enough large range of variables to cover all possible situations. Thus, we
can know the mathematical odds of one of more of your opponents holding a
certain card formation and a formation higher than yours, at every stage
of the game.
The
results were gathered and the outcome is the book (Draw Poker Odds: The
Mathematics of Classical Poker) (see the Books
section) and a software (Draw Poker Odds
Calculator 1.1).
The
software is based on the formulas from the book and is the first poker
odds calculator that uses compact probability formulas instead of partial
simulations. See the software section for more
details.
The
draw poker
most often practiced is one with 52 cards, without wild cards,
and with the ability to discard up to five cards. We will present the
probabilities involved in this type of game.
We are interested only in the card distributions
that generate the predicted events. Additional information like the way
your opponents discard or their psychological profile is not taken into
account.
The following types of probabilities are
calculated:
– Initial probabilities on the first card
distribution for your own hand;
– Prediction probabilities after first card
distribution and before the second for your own hand;
– Prediction probabilities for opponents’
hands.
The
events whose probability is to be measured are the occurrence of the
various combinations of cards (of size 1, 2, 3, 4 or 5) in your own hand
or your opponents’ hands.
1)
Initial probabilities on the first card distribution for your own
hand
In
this section we present the probabilities of receiving various valuable
formations from the first five cards dealt.
Event

P
(odds)

P
(percent)

one
pair

0.73
: 1

42.256

two
pair

20
: 1

4.753

three
of a kind

47
: 1

2.112

straight

256
: 1

0.394

flush

526
: 1

0.198

full
house

693
: 1

0.144

four
of a kind

5000
: 1

0.024

straight
flush

100000
: 1

0.001

at
least one pair

1.15
: 1

53.637

at
least two pair

18
: 1

5.186

at
least three of a kind

42
: 1

2.352

at
least three consecutive cards

0.47
: 1

32.288

at
least three suited cards

0.59
: 1

37.106

In
this table, "at least one pair" means you can also achieve two
pair, three of a kind, a full house or four of a kind; "at least two
pair" means you can also achieve three of a kind, a full house or
four of a kind and the like.
2)
Prediction probabilities after first card distribution and before
the second for your own hand
These
are the probabilities of receiving various card combinations at the second
distribution, after you have been dealt the first five cards and you have
discarded (or not).
There
are 17 cases of possible types of dealt combinations that require making a
decision to keep or discard cards the most (one pair + high card, one pair
+ three suited, one pair + three from a straight, ...., two pair + three
suited, ..., three from a straight + three suited, four suited + four from
a straight, etc.)
All these cases are presented in detail along
with the associated odds and conclusions in the book Draw Poker Odds.
We present here only one of these cases:
5)
One pair + three suited (the suit does not contain any
paired card) (example:
7♥ 7♠ 8♣ J♣ K♣)
The player has two playing options to achieve a
valuable formation: holding only the pair and discarding the suited cards,
or holding the suited cards and discarding the paired cards.
A)
Held: (77) Discarded:
(8♣ J♣ K♣) Goal:
two pair, three of a kind, full house or four of a kind
The probabilities of achieving these target
formations are
 two pair:
15.98519%
 three of a kind:
11.43385%
 full house:
1.01757%
 four of a kind:
0.24668%,
totaling 28.68329%.
B)
Held: (8♣ J♣ K♣) Discarded: (77)
Goal: flush
The number of all possible combinations for the
second distribution is C(47, 2) = 1081.
The favorable combinations are (♣♣),
numbering C(133, 2) = 45, so the probability of achieving a flush is
P = 45/1081 = 0.0416281 = 4.16281%.
Let us also calculate the probability of
achieving three of a kind in subcase B:
The favorable combinations are (xx), x
being any of the held cards, numbering 3C(3, 2) = 9.
The probability of achieving three of a kind is
P = 9/1081 = 0.0083256 = 0.83256%.
Conclusions:
– The probability of achieving a flush in subcase B is
lower than the probability of achieving three of a kind or better in
subcase A and has a low value itself.
– The probability of achieving three of a kind in subcase B
is lower than the probability of achieving three of a kind in subcase A.
Therefore, holding the pair and discarding the
suited cards is recommended with respect to the goal of achieving the
target formations.
3)
Prediction probabilities for opponents’ hands
These
are the probabilities for one or many of your opponents to hold various
card formations at any moment of the game, function of the value or/and
symbol distributions of your own seen cards (the first five dealt plus
their replacements).
We present here only the probabilities of one or
more opponents holding three of a kind as a final (or initial)
hand, as a function of the information given by your own seen cards (the
first five dealt plus their replacements).
The set of parameters the formula depends on are:
n = number
of opponents (n could be 1, 2, 3, or 4);
c = number
of seen cards (c could be 5, 6, 7, 8, 9 or 10);
We
are not interested in how many cards of each value in a distribution, but
in a cumulative representation: how many values are represented by four
cards, how many by three, and so on, how many by zero. This is the
hypothesis for any calculus of the probability of the occurrence of a card
formation made of values in your opponents’ hands.
We call value distribution any finite
sequence of numbers of values from the seen cards, in the cumulative form.
For example, if the seen cards are 2♣
3♠ 3♥ 5♦ 5♣ 5♥ 8♦ J♠ Q♥,
their value distribution is 3211110000,
meaning there are six represented values, in this way: three cards of one
value (5), two cards of another value (3), one card of another value (2),
one card of another value (8), one card of another value (J) and one card
of another value (Q); the rest of the values (4, 6, 7, 9, 10, K, A) are
not represented.
As a convention, we remove the zeroes from
this denotation, so the above value distribution is denoted by
321111.
If a maximum number of five replacements is
allowed (c could be maximum 10), there are 82 possible value
distributions.
The next table notes the probabilities of one and
at least one of your opponents holding a three of a kind formation,
corresponding to the value distribution of your own seen cards.
The numerical values are returned by the formula
of number of favorable card combinations for one oponent to hold three
of a kind and some formulas of a classical probability repartition.
The existence of a dash in the column n = 5
indicates that such case is impossible (if fivecard replacement is alowed
– so c could be up to 10, the maximum number of players can be
maximum five, so you may play against a maximum of four opponents).
Probabilities of opponents holding three of a kind
Distribution

n
= 1

n
= 2

n
= 3

n
= 4

n
= 5

41

2.46685%

4.87285%

7.21950%

9.50826%

11.74056%

32

2.41522%

4.77211%

7.07207%

9.31649%

11.50670%

311

2.31339%

4.57326%

6.78086%

8.93739%

11.04403%

221

2.25850%

4.46599%

6.62363%

8.73254%

10.79382%

2111

2.15621%

4.26594%

6.33017%

8.34989%

10.32607%

11111

2.05367%

4.06516%

6.03534%

7.96507%

9.85517%

42

2.56793%

5.06992%

7.50766%

9.88280%

12.19696%

411

2.45996%

4.85941%

7.19983%

9.48268%

11.70938%

33

2.57114%

5.07617%

7.51680%

9.89468%

12.21142%

321

2.40466%

4.75150%

7.04191%

9.27724%

11.45882%

3111

2.29604%

4.53936%

6.73117%

8.87266%

10.96498%

222

2.34615%

4.63727%

6.87463%

9.05949%

11.19311%

2211

2.23731%

4.42456%

6.56288%

8.65337%

10.69708%

21111

2.12817%

4.21105%

6.24961%

8.24478%

10.19750%

111111

2.01874%

3.99673%

5.93480%

7.83374%

9.69434%

43

2.73704%

5.39916%

7.98843%

10.50682%

12.95629%

421

2.56024%

5.05494%

7.48577%

9.85436%

12.16232%

4111

2.44492%

4.83006%

7.15689%

9.42683%

11.64128%

331

2.56360%

5.06148%

7.49533%

9.86678%

12.17744%

322

2.50131%

4.94006%

7.31781%

9.63609%

11.89638%

3211

2.38558%

4.71425%

6.98736%

9.20626%

11.37222%

31111

2.26951%

4.48752%

6.65520%

8.77368%

10.84408%

2221

2.32288%

4.59180%

6.80803%

8.97277%

11.08723%

22111

2.20657%

4.36446%

6.47473%

8.53844%

10.55661%

211111

2.08994%

4.13620%

6.13970%

8.10132%

10.02195%

1111111

1.97298%

3.90703%

5.80292%

7.66141%

9.48323%

44

2.91711%

5.74912%

8.49852%

11.16772%

13.75906%

431

2.73294%

5.39120%

7.97681%

10.49176%

12.93798%

422

2.66665%

5.26219%

7.78851%

10.24748%

12.64087%

4211

2.54363%

5.02256%

7.43843%

9.79286%

12.08740%

41111

2.42024%

4.78191%

7.08642%

9.33516%

11.52947%

332

2.67033%

5.26936%

7.79898%

10.26106%

12.65740%

3311

2.54713%

5.02938%

7.44840%

9.80582%

12.10319%

3221

2.48028%

4.89904%

7.25781%

9.55808%

11.80130%

32111

2.35643%

4.65733%

6.90402%

9.09776%

11.23982%

311111

2.23221%

4.41460%

6.54827%

8.63432%

10.67380%

2222

2.41324%

4.76825%

7.06643%

9.30914%

11.49774%

22211

2.28912%

4.52584%

6.71136%

8.84685%

10.93346%

221111

2.16462%

4.28240%

6.35433%

8.38141%

10.36462%

2111111

2.03976%

4.03792%

5.99533%

7.91281%

9.79117%

11111111

1.91453%

3.79242%

5.63435%

7.44102%

9.21310%

441

2.91711%

5.74912%

8.49852%

11.16773%



432

2.85062%

5.61998%

8.31040%

10.92413%



4311

2.71952%

5.36508%

7.93869%

10.44232%



4221

2.64825%

5.22637%

7.73622%

10.17960%



42111

2.51642%

4.96952%

7.36089%

9.69208%



411111

2.38417%

4.71151%

6.98335%

9.20104%



333

2.85477%

5.62805%

8.32217%

10.93937%



3321

2.65209%

5.23385%

7.74715%

10.19379%



33111

2.52006%

4.97661%

7.37125%

9.70556%



3222

2.58052%

5.09444%

7.54350%

9.92937%



32211

2.44817%

4.83640%

7.16617%

9.43890%



321111

2.31540%

4.57719%

6.78662%

8.94489%



3111111

2.18222%

4.31682%

6.40484%

8.44730%



22221

2.37607%

4.69568%

6.96019%

9.17088%



222111

2.24299%

4.43568%

6.57918%

8.67461%



2211111

2.10950%

4.17450%

6.19594%

8.17475%



21111111

1.97559%

3.91215%

5.81046%

7.67127%



111111111

1.84127%

3.64863%

5.42272%

7.16415%



442

3.04702%

6.00119%

8.86536%

11.64226%



4411

2.90736%

5.73020%

8.47097%

11.13206%



433

3.05172%

6.01031%

8.87862%

11.65940%



4321

2.83565%

5.59090%

8.26802%

10.86923%



43111

2.69494%

5.31726%

7.86891%

10.35180%



4222

2.75924%

5.44236%

8.05144%

10.58853%



42211

2.61818%

5.16781%

7.65069%

10.06857%



421111

2.47664%

4.89195%

7.24744%

9.54460%



4111111

2.33464%

4.61477%

6.84167%

9.01659%



3331

2.84000%

5.59935%

8.28034%

10.88519%



3322

2.76348%

5.45059%

8.06344%

10.60409%



33211

2.62217%

5.17559%

7.66206%

10.08333%



331111

2.48040%

4.89929%

7.25817%

9.55855%



32221

2.54506%

5.02535%

7.44251%

9.79816%



322111

2.40294%

4.74813%

7.03698%

9.27083%



3211111

2.26034%

4.46959%

6.62891%

8.73942%



31111111

2.11728%

4.18973%

6.21830%

8.20393%



22222

2.46771%

4.87452%

7.22195%

9.51145%



222211

2.32523%

4.59640%

6.81476%

8.98154%



2221111

2.18228%

4.31695%

6.40503%

8.44755%



22111111

2.03887%

4.03617%

5.99275%

7.90944%



211111111

1.89498%

3.75406%

5.57790%

7.36719%



1111111111

1.75062%

3.47060%

5.16047%

6.82076%



Example
of using the table:
Assume you are playing against three opponents, you have been dealt
(77JJK), you hold (77JJ), discard (K) and replace it with (3).Your final
hand is two pair: (77JJ3).
You want the probability for at least one
opponent to hold three of a kind.
You
have c = 6 seen cards, with the value distribution 2211 (two J's,
two 7's, one K and one 3).
By looking in the table at the intersection of
row 2211 with column n = 3, you find the probability
6.56288%.
Of
course, this is not the probability that one or more opponents hold a
higher formation than yours (there are also the straights, flushes, full
houses, four of a kind and also the higher two pairs to count, but these
can be also calculated).
Sources 
In the book Draw Poker Odds: The Mathematics of Classical Poker
you will find sections for the probabilities of your opponents holding:
one pair, two pair, three of a kind, straight, flush, full house, four of
a kind, straight flush, higher one pair, higher two pair, higher three of
a kind, higher straight, higher flush, higher full house, higher four of a
kind, higher straight flush and the overall probability of one and at
least one opponent holding a higher formation than yours. And all these
for any stage of the game. You will also find all the used formulas, along
with the way of computing them and the supporting math. See the Books
section for details.
If you want to skip the math and use the
numerical results directly, you may use Draw Poker Odds Calculator
1.1 to search for odds at home or while playing on line. See
the Software section for details. 

